Abstract—The problem of adaptive stochastic airline seat inventory control lies at the heart of airline revenue management. This problem concerns the allocation of the finite seat inventory to the stochastic customer demand that occurs over time before the flight is scheduled to depart. The objective is to find the right combination of customers of various fare classes on the flight such that revenue is maximized. In this paper, the unbiased static and dynamic policies of stochastic airline seat inventory control (airline booking) are developed under parametric uncertainty of underlying models, which are not necessarily alternative. For the sake of simplicity, but without loss of generality, we consider (for illustration) the case of nonstop flights with two fare classes. The system developed is able to recognize a situation characterized by the number of reservations made by customers of the above fare classes at certain moment of time before departure. The proposed policies of the airline seat inventory control are based on the use of order statistics of cumulative customer demand, which have such properties as bivariate dependence and conditional predictability. Dynamic adaptation of the system to airline customer demand is carried out via the bivariate dependence of order statistics of cumulative customer demand. Dynamic optimization of the airline seat allocation is carried out via the conditional predictability of order statistics. The system makes on-line decisions as to whether to accept or reject any customer request using established decision rules based on order statistics of the current cumulative customer demand. The computer simulation results are promising.
Index Terms—Airline booking, dynamic adaptation and optimization, stochastic demand
I. INTRODUCTION
ASSENGER reservation systems have evolved from low level inventory control processes to major strategic information systems. Today, airlines and other transportation companies view revenue management systems and related information technologies as critical determinants of future success. Indeed, expectations of revenue gains that are possible with expanded revenue management capabilities are now driving the acquisition of new information technology.
Manuscript received March 10, 2013; revised April 06, 2013. This work was supported in part by Grant No. 06.1936, Grant No. 07.2036, and Grant No. 09.1014 from the Latvian Council of Science and the National Institute of Mathematics and Informatics of Latvia.
Nicholas. A. Nechval is with the Statistics Department, EVF Research Institute, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
Konstantin N. Nechval is with the Applied Mathematics Department, Transport and Telecommunication Instutute, Riga LV-1019, Latvia (e-mail: [email protected]).
Maris Purgailis is with the Cybernetics Department, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
Uldis Rozevskis is with the Informatics Department, University of Latvia, Riga LV-1050, Latvia (e-mail: [email protected]).
Each advance in information technology creates an opportunity for more comprehensive reservations control and greater integration with other important transportation planning functions. It is common practice for airlines to sell a pool of identical seats at different prices according to different booking classes to improve revenues in a very competitive market. In other words, airlines sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions. The question then arises whether to offer seats at a relatively low price at a given time with a given number of seats remaining or to wait for the possible arrival of a higher paying customer. Assigning seats in the same compartment to different fare classes of customers in order to improve revenues is a major problem of airline seat inventory allocation. This problem has been considered in numerous papers. For details, the reader is referred to a review of yield management, as well as perishable asset revenue management, by Weatherford et al. [1], and a review of relevant mathematical models by Belobaba [2].
This paper deals with the airline seat allocation problem when customers for different fare levels are booked into a common seating pool in the aircraft. The following assumptions are made: (1) single-leg flight: bookings are made on the basis of a single departure and landing; no allowance is made for the possibility that bookings may be part of larger trip itineraries, (2) independent demands: the demands for different fare classes are stochastically independent, (3) low before high demands: the lowest fare reservations requests arrive first, followed by the next lowest, etc., (4) no cancellations: cancellations, no-shows and overbooking are not considered, (5) nested classes: any fare class can be booked into seats not taken by bookings in lower fare classes, (6) fare classes: the business and economy fare classes are considered.
The first purpose of this paper is to present the innovative information technologies for constructing the unbiased static and dynamic policies of the airline seat inventory allocation on the basis of the ‘unbiasedness performance index’. The static and dynamic policies (unbiased) are more efficient (from the point of view of airline revenue management) as compared with the policies, where the unknown parameters of the airline customer demand models are estimated and then treated as if they were the true values. At the initial stage of airline booking it may be used the static policy of seat inventory allocation, and at the fundamental stage may be used the dynamic policy.
The second purpose of this paper is to introduce the idea of prediction of a future cumulative customer demand for the seats on a flight via the order statistics from the underlying
Adaptation and Optimization of Stochastic
Airline Seat Inventory Control
Nicholas A. Nechval, Member, IAENG, Konstantin N. Nechval, Maris Purgailis, and Uldis Rozevskis
distribution, where only the functional form of the distribution is specified, but some or all of its parameters are unspecified. This idea allows one to use the technique of invariant embedding of sample statistics in a performance index in order to eliminate the unknown parameters from the problem [3-5]. The technique represents a simple and computationally attractive statistical method based on the constructive use of the invariance principle in mathematical statistics. Unlike the Bayesian approach, an invariant embedding technique is independent of the choice of priors, i.e., subjectivity of investigator is eliminated from the problem. It allows one to find the improved invariant statistical decision rules, which have smaller risk than any of the well-known traditional statistical decision rules, and to use the previous and current sample data as completely as possible.
II. STATE-OF-THE-ART AND PROGRESS BEYOND Airline seat allocation is a very profitable tool in the airline industry. A major problem of airline seat allocation is to sell the same seat at different prices according to different types of travelers (first class, business and economy) and other conditions in order to improve revenues. This problem has been considered in numerous papers. Littlewood [6] was the first to propose a solution method of the airline seat allocation problem for a single-leg flight with two fare classes. The idea of his scheme is to equate the marginal revenues in each of the two fare classes. He suggests closing down the low fare class when the certain revenue from selling low fare seat is exceeded by the expected revenue of selling the same seat at the higher fare. That is, low fare booking requests should be accepted as long as
{
}
,Pr 1 1 1
2 c Y u
c ≥ > (1)
where c1 and c2 are the high and low fare levels respectively, Y1 denotes the demand for the high fare (or business) class, u1 is the number of seats to protect for the high fare class
and Pr{Y1>u1} is the probability of selling more than u1
protected seats to high fare class customers. The smallest value of u1 that satisfies the above condition is the number of
seats to protect for the high fare class, and is known as the protection level of the high fare class customers. The concept of determining a protection level for the high fare class can also be seen as setting a booking limit, a maximum number of bookings, for the low fare class. Both concepts restrict the number of bookings for the low fare class in order to accept bookings for the high fare class.
It should be remarked that there is no protection level for the low fare (or economy) class; u2 is the booking limit, or
number of seats available, for the low fare class; the low fare class is open as long as the number of bookings in this class remains less than this limit. Thus, (u1+u2) is the booking
limit or number of seats available for the high fare class at time. The high fare class is open as long as the number of bookings in this and low classes remain less than this limit.
Richter [7] gave a marginal analysis, which proved that (1) gives an optimal allocation (assuming certain continuity conditions). Optimal policies for more than two classes have been presented independently by Curry [8], Wollmer [9],
Brumelle & McGill [10], and Nechval et al. [11].
III. AIRLINE BOOKING POLICIES KNOWN FROM PRACTICE
A. Static Airline Booking Policy under Certainty
It will be noted that (1) represents the static policy of airline seat allocation (or airline booking) under complete information. If Fθ, the probability distribution function of Y1
with the parameter θ (in general, vector), is continuous and strictly increasing, the definition (1) of u1 is equivalent to
) ) ( (
arg 1
1= Fθ u =γ
u (2)
where
, / 1 2 c c
=
γ (3)
). ( 1 )
(u1 F u1
Fθ = − θ (4)
B. Static Airline Booking Policy under Uncertainty
In practice, under parametric uncertainty, i.e. when the parameter θ is unknown, the performance index,
, ) ( 1 γ
θ u =
F) (5)
is usually used to construct the static policy given by ),
) ( (
arg 1
1= Fθ u =γ
u )
(6) whereθ)represents the maximum likelihood estimator of θ. The performance index (5) is named as ‘maximum likelihood performance index’. The static policy (6) based on (5) is named as ‘static maximum likelihood airline
booking policy’.
C. Dynamic Airline Booking Policy
The static policy of airline booking is optimal as long as no change in the probability distributions of the customer demand is foreseen. However, information on the actual customer demand process can reduce the uncertainty associated with the estimates of demand. Hence, repetitive use of a static policy over the booking period, based on the most recent demand and capacity information, is the general way to proceed.
IV. AIRLINE BOOKING POLICIES PROPOSED IN THE PAPER
A. Static Airline Booking Policy under Uncertainty
This policy is based on the performance index, ,
} ) ( { θ 1 γ
θ F u =
E (7)
which takes into account (2) and the previous data of cumulative customer demand Y1 for the seats on a flight. It
allows one to construct the static unbiased airline booking policy given by
), } ) ( { (
arg 1
) unb (
1 = Eθ Fθ u =γ
u (8)
where u1≡u1(θ)),θ)represents either the maximum likelihood estimator of θ or sufficient statistic S for θ, i.e.,
). (
1
1 u S
booking policy’.
The relative bias of the static airline booking policy is given by
%. 100 | } ) ( { | )
( 1 γ1
γ
θ
θ −
= E F u
u
r (9)
B. Dynamic Airline Booking Policy under Certainty In this section, we consider a flight for a single departure
date with m predefined reading dates at which the dynamic policy is to be updated, i.e., the booking period before departure is divided into m readings periods: (0, τ1], (τ1, τ2],
…, (τm-1, τm] determined by the m reading dates: τ1, τ2, …,
τm. These reading dates are indexed in increasing order: 0<τ1<τ2< ⋅⋅⋅ <τm, where (τm-1,τm] denotes the reading period immediately preceding departure, and τm is at departure. Typically, the reading periods that are closer to departure cover much shorter periods of time than those further from departure. For example, the reading period immediately preceding departure may cover 1 day whereas the reading period 1-month from departure may cover 1 week.
Let us suppose that the cumulative passenger demand for the high fare class at the kth reading date (time τk, 1≤k≤m) is Y1k representing the kth order statistic from the underlying
distribution with the probability distribution function Gθ (y1k), where θ is a parameter (in general, vector). In other
words, Y1k represents the number of seats sold for the
customers of the high fare class at the kth reading date. We assume that the cumulative passenger demands for the high and low fare classes are stochastically independent. Each booking of a seat of the high fare class generates average revenue of c1. Each booking of a seat of the low fare class
generates average revenue of c2, where c2<c1. Let u1k be an
individual protection level for the high fare class at time τk (the kth reading date). This many seats are protected for the high fare class from the low fare class. There is no protection level for the low fare class; u2k is the booking
limit for the low fare class at time τk; the low fare class is open as long as the number of bookings in this class remains less than this limit. Thus, (u1k+u2k) is the booking limit for
the high fare class at time τk. The high fare class is open as long as the number of bookings in this and low classes remain less than this limit. The maximum number of seats that may be booked by fare classes in the next at time τk prior to flight departure is the number of unsold seats uok.
Under the complete information, the dynamic airline booking policy is given by
, 1 ..., , 2 , 1 ), ) | ( (
arg 1 1
1 = G u y = k = m−
uk θ k k γ (10)
where
Gθ(u1k|y1k)=1−Gθ(u1k|y1k), (11) )
| (u1k y1k
Gθ represents the conditional probability distribution function of the mth order statistic Y1m. The
number of unsold seats protected for the high fare class from the low fare class in the next at time τk prior to flight departure is the number of unsold seats, u1ok, which is given by
). ,
min( 1 1
1k uk uk yk
uo = o − (12)
C. Dynamic Airline Booking Policy under Uncertainty
Under the parametric uncertainty, the dynamic unbiased airline booking policy is given by
, 1 ..., , 2 , 1 ), )} | ( { (
arg 1 1
) unb (
1 = E G u y = k= m−
uk θ θ k k γ (13)
where u1k ≡u1k(θ)),θ)represents either the maximum likelihood estimator of θ or sufficient statistic S for θ, i.e.,
). (
1
1 u S
uk ≡ k The number of unsold seats protected for the high fare class from the low fare class in the next at time τk prior to flight departure is the number of unsold seats, which is given by
min( , 1 ).
) unb ( 1 )
unb (
1k uk uk yk
uo = o − (14)
V. MATHEMATICAL PRELIMINARIES
Theorem 1. Let X1 ≤ ... ≤ Xk be the first k ordered observations (order statistics) in a sample of size m from a continuous distribution with some probability density function fθ(x) and distribution function Fθ(x), where θ is a parameter (in general, vector). Then the joint probability density function of X1≤ ... ≤ Xk and the lth order statistics Xl (1 ≤ k < l ≤ m) is given by
), | ( ) ..., , ( ) , ..., ,
(x1 xk xl g x1 xk g xl xk
gθ = θ θ (15)
where
) ,..., (x1 xk
gθ ( )[1 ( )] ,
)! (
!
1
∏
=−
− −
= k
i
k m k
i F x
x f k m
m
θ
θ (16)
1
) ( 1
) ( ) ( )! ( )! 1 (
)! ( ) | (
− −
− − −
− −
− =
k l
k k l
k l
x F
x F x F l m k l
k m x
x g
θ θ θ
θ
) ( 1
) ( )
( 1
) ( ) ( 1
k l l m
k k l
x F
x f x
F x F x F
θ θ θ
θ θ
−
− − −
×
−
∑
−−
=
− −
− − −
−
= 1
0
1 )!
( )! 1 (
)!
( l k
j j
k l l m k l
k m
) ( 1
) ( )
( 1
) ( 1 ) 1 (
k l j l m
k l j
x F
x f x
F x F
θ θ θ
θ
−
− − − ×
+ −
∑
−=
−
− − −
−
= ml
j j
l m l m k l
k m
0
)! ( )! 1 (
)! (
) ( 1
) ( )
( 1
) ( ) ( ) 1 (
1
k l j k l
k k l
j
x F
x f x
F x F x F
θ θ θ
θ θ
−
− − −
×
+ − −
(17)
represents the conditional probability density function of Xl given Xk=xk.
Proof. The joint density of X1≤ ... ≤ Xk and Xl is given by
) , ..., , (x1 xk xl
gθ
∏
=
− − −
= k
i i
x f l m k l
m
1
) ( )! ( )! 1 (
!
l m l l k l k
l F x f x F x
x
F − −− − −
×[ θ( ) θ( )] 1 θ( )[1 θ( )]
=gθ(x1 ,...,xk)gθ(xl|xk). (18) It follows from (18) that
), | ( ) ..., , ( ) , ..., , ( ) ..., , | ( 1 1
1 l k
k l k k
l g x x
x x g x x x g x x x g θ θ θ
θ = =
(19)
i.e., the conditional distribution of Xl, given Xi = xi for all i = 1, …, k, is the same as the conditional distribution of Xl , given only Xk = xk, which is given by (17). This ends the proof.
Corollary 1.1. The conditional probability distribution
function of Xl given Xk=xk is
{
}
)! ( )! 1 ( )! ( 1 | l m k l k m x X x XP l l k k − − −
− − = = ≤ θ j l m k l k l j j x F x F j l m j k
l −++
− − = − − + + − − − − ×
∑
1 10 1 ( )
) ( 1 1 ) 1 ( 1 θ θ j k l j l m l m k l k
m m l j
j − +
− − − − − − =
∑
− = ) 1 ( )! ( )! 1 ( )! ( 0 . ) ( 1 ) ( )( l k j
k k l x F x F x
F − +
− − θ θ θ (20)
Corollary 1.2. Let X1 ≤ ... ≤ Xk be the first k order statistics in a sample of size m from the two-parameter Weibull distribution with the probability density function
), 0 ( exp ) ( 1 > − = − x x x x f δ δ
θ βδ β β (21)
where θ = (β,δ), β>0 and δ>0 are the scale and shape parameters, respectively. Then the conditional probability distribution function of Xl given Xk=xk is
{
}
)! ( )! 1 ( )! ( 1 | l m k l k m x X x XP l l k k
− − − − − = = ≤ θ . exp 1 ) 1 ( 1 1 1 0 j l m k l k l j j x x j l m j k
l −++
− − = − − + + − − − −
∑
δ δ δ β (22)Theorem 2. If in (22) the scale parameter β is unknown, then the predictive probability distribution function of Xl based on (xk,δ ) is given by
)! ( )! 1 ( ! 1 l m k l m x x X X P k l k l − − − − = ≤
δ δ
δ
∑
− − = − + + − − − × 1 0 1 ) 1 ( 1 k l j j j l m j k l . ) 1 ( ) 1 ( 1 1 1 0 − − = + + − + + + − − ×
∏
ks k
l m l j m k s
x
x δ
(23)
Proof. We reduce (22) to
= ≤
δ δ δ δ
θ βk βk
k l
k
l X x
x x X X P
∑
− − = − − − − − − − = 1 0 1 )! ( )! 1 ( )! ( 1 k l j j k l l m k l k m[
]
m l jj v w j l m + + − − − + + − −
× exp( [ 1]) 1
1 ) 1
( δ
{
V v |W w}
,P ≤ =
= δ δ δ (24)
where V = Xl / Xk is the ancillary statistic whose distribution does not depend on the parameter β. Since Xk does not depend on V, W =(Xk/β)δis the pivotal quantity, whose distribution is known and does not depend on the parameters
β and δ, we eliminate the parameter β from the problem as
, ) ( } | { } { 0 k k k k l l l
l x P X x X x g x dx
X
Pδ
∫
θ θ∞ = ≤ = ≤ (25) where ) ( )! ( )! 1 ( ! )
( 1 k
k
k F x
k m k m x g − − − = θ θ ), , 0 ( ), ( )] ( 1 [ − ∈ ∞
× − k k
k m
k f x x
x
Fθ θ
(26) represents the probability density function of the kth order statistic Xk. Indeed, it follows from (26) that
1 exp 1 )! ( )! 1 ( ! ) ( − − − − − = k k k k x k m k m dx x g δ θ β δ δ δ β β
β
− − × − k k k m k x d x x exp exp ) ( . ) ( ] 1 [ )! ( )! 1 (
! 1 ( 1)
dw w g dw e e k m k
m − w k wm k =
− −
= − − − −+ (27)
It follows from (24) and (27) that
∫
∞ = ≤ = ≤ 0 ) ( } | { }{V v P V v W w g wdw
Pδ δ δ δ δ δ
∑
− − = − + + − − − − − − − = 1 0 1 ) 1 ( 1 )! ( )! 1 ( ! 1 k l j j j l m j k l l m k l m( )
[
1( 1 ) ( 1 )]
.1 1 0 − − = + + − + + + − −
×
∏
ks s k m j l m
vδ (28)
Now (23) follows from (28). This ends the proof.
Corollary 2.1. If the parameter δ =1, i.e. we deal with
∑
− − = − + + − − − − − − − = ≤ 1 0 1 ) 1 ( 1 )! ( )! 1 ( ! 1 k l j j k l k l j l m j k l l m k l m x x X X P . ) 1 ( ) 1 ( 1 1 1 0 − − = + + − + + + − −×
∏
ks k
l m l j m k s
x x
(29)
Theorem 3. Let X1 ≤ ... ≤ Xk be the first k ordered observations from a sample of size m from the
two-parameter Weibull distribution (21). Then the joint probability density function of the pivotal quantities
2 ,
δ δ
)
=
W 3 ,
δ
β β) )
= W (30)
conditional on fixed z(k)=(zi, …, zk), where , ..., , 1 , ) /
(X i k
Zi = i β δ =
)
)
are ancillary statistics, any k−2 of which form a functionally independent set,β) and δ)are the estimators of β and δ, based on the first k ordered
observations (X1≤ ... ≤Xk) from a sample of size m from the two-parameter Weibull distribution (21), such that W2 and W3 are the pivotal quantities (in particular, the maximum
likelihood estimators of β and δ,
δ δ δ β ) ) )
) 1/
1 ) ( − + =
∑
= k x k m x k i k i (31) and 1 1 1 1 1 ln 1 ) ( ln ) ( ln − = − = = − − + − +=
∑
∑
∑
ki i k i k i k k k i i i x k x k m x x x k m x
xδ δ δ δ
δ) ) ) ) ) (32)
respectively, lead to the pivotal quantities W2 and W3) is
given by 1 3 1 1 2 ) ( ) ( 3
2, | ) ( ) 2 2
( −
= −
•
∏
= k kw
i w i k k k w z w w w
f z ϑ z
− + − ×
∑
= k i w k wi m k z
z ww
1
32 2 ( ) 2
exp ) 1 ( 3 1 2 2 )
( ) 2 2
( −
= −
•
∏
= k w k
i w i k k w z w z ϑ 1 3 2 1
32 2 ( ) 2 2
exp − = − + −
×
∑
k wi w k w i w w w z k m z w ), | ( ) |
( 3 2 ( )
) ( 2 k k w w f w
f z ,z
= w2∈(0, ∞), w3∈(0, ∞), (33)
where
1
2
0 1 1
2 2 )
( ) ( ) 2 2 ( ) 2
( − ∞ − = = − • − + Γ
=
∫
k w∏
z∑
z m k z dwk k i w k w i k i w i k k z ϑ (34) is the normalizing constant,
, ) ( ) ( ) | ( 1 1 2 2 ) ( ) (
2 2 2 2
k k i w k w i k i w i k k k z k m z z w w f − = = − − +
= z
∏
∑
z ϑ
), , 0 (
2∈ ∞
w (35) , ) ( ) ( 1 2
0 1 1
2 2 )
( 2 2 2
− ∞ − = = − − +
=
∫
w∏
z∑
z m k z dwk k i w k w i k i w i k k z ϑ (36) ) 1 ( 3 1 ) ( 2 3 2 2 2 ) ( ) ( ) , | ( = − Γ − + =
∑
k w k k i w k w i k w k z k m z w w f z ). , 0 ( , ) (exp 2 3 1 3
1
32 2 2 2 ∈ ∞
− + − × − =
∑
z m kz ww ww w k i w k w i w (37)
Proof. The joint density of X1≤ ... ≤ Xk is given by
∏
= − − = k i i k x k m m x x f 1 1 1 )! ( ! ) ..., , ( δθ βδ β
. ) ( exp
exp
− − − × δ δ β
βi k
x k m x
(38)
Using β) and δ)(the maximum likelihood estimators of β and δ obtained from solution of (31) and (32)) and the invariant embedding technique [3-5], we transform (38) as follows: k k i i k x k m m d d x x
fθ β δ
∏
δ= − − = 1 1 1 )! ( ! ) ..., , ( ) ) δ β β β β δ δ δ ) ) d d x k m x x k i k k i i
i exp ( )
1 1
∏
∑
= = − − − × ) 1 ( 3 1 2 2 11 2 2
)! ( ! − = − = −
∏
∏
− −= k w k
i w i k k i i k w z w x k m
m β)δ)
. )
(
exp 2 3
1 3 2 1
32 z 2 m k z 2 w w 2 dw dw
w w k i w k w i w − = − + −
×
∑
(39)Normalizing (39), we obtain (33). This ends the proof.
Theorem 4. If in (8) both parameters β and δ are unknown, then the predictive probability distribution function of Xl based on ( ,δ)
)
k
x and conditional on fixed z(k)
is given by
)! ( )! 1 ( ! 1 ) ( l m k l m x x X X P k k l k l − − − − = ≤ z δ
δ) )
∑
∫
−− = ∞ + + − − − − × 1 0 0 1 ) 1 ( 1 k l j j j l m j k l . ) | ( ) 1 ( ) 1 (1 2 () 2
1 1 0 2 dw w f s k m j l m x x k k s w k l z − − = + + − + + + − − ×
∏
δ) (40)Proof. We reduce (23) to
{
}
) / ( ) /( 2 2
2 2 w W k l k
l X x x P V v
X
P = ≤
∑
−−
= − + +
−
− −
− − − −
= 1
0 1
) 1 ( 1 )!
( )! 1 (
! 1
k l
j
j
j l m j k l l m k l
m
, )] 1 ( ) 1 )( 1 [(
1 1
0 22
− −
=
+ + − + + + − −
×
∏
ks w
s k m j l m v
(41)
where δ
)
) / (
2 Xl Xk
V = is the ancillary statistic whose distribution does not depend on the parameters β and δ. Since the pivotal quantity W2, whose distribution is given by
(35), does not depend on V2, it follows from (41) and (35)
that
{
|}
{
}
( | ) 2,) ( 2 0
2 2 )
( 2
2 v P V 2 v 2 f w dw
V
P zk
∫
W w zk∞
≤ =
≤ (42)
where the unknown parameters β and δ are eliminated from
the problem. Now (40) follows from (42). This ends the proof.
VI. ILLUSTRATIVE EXAMPLE OF AIRLINE BOOKING POLICIES Let X1, …, Xn be the random sample of the previous independent observations of the cumulative customer demand for the high fare class, which follow the exponential distribution with the probability density function (21) (δ=1), where the parameter β is unknown. Then the static policies of airline booking under parametric uncertainty are given as follows.
The static maximum likelihood airline booking policy
follows from (6):
,
ln /
) ml ( 1
n S
u = γ− (43)
where =
∑
n= i XiS
1 is the sufficient statistic for β, with
, 0 ), exp( )
( 1 ) ( ~
/ 1 − ≥
Γ =
= v − v v
n v f S
V β n (44)
and the relative bias,
%. 100 | ) ln 1 ( | % 100 | } ) ( { | ) (
1 / 1 )
ml ( 1 )
ml (
1 γ
γ γ
γ γ
θ
θ − = + −
= E F u − n −
u r
(45) If, say, n=1 and γ=0.4, then rrb(u1(ml))=30%. Thus, in this example the static maximum likelihood airline booking policy has the relative bias equal to 30%. It follows that the protection level for customers of the high fare class will be determined incorrectly. This may lead to serious loss.
The static unbiased airline booking policy follows from
(8):
[ 1] ,
/ 1 ) unb (
1 S
u = γ− n− (46)
where the relative bias r(u1(unb))=0.
The dynamic unbiased airline booking policy follows
from (13) and (29):
, )
1 ( ) 1 ( 1
1 ) 1 ( 1 )!
1 (
!
arg 1
1
0 1
1 1
0 )
unb ( 1
=
+ + − + +
− ×
+ −
− −
− −
= −
−
=
− −
=
∏
∑
γ
k
s k
k k m
j
j
k
s k m j y
u
j j
k m k
m m
u
k=1,2,...,m−1. (47)
min( , 1 ).
) unb ( 1 )
unb (
1k uk uk yk
uo = o − (48)
VII. CONCLUSION AND DIRECTIONS FOR FUTURE RESEARCH In this paper, we develop a new frequentist approach to improve predictive statistical decisions for airline seat allocation problems. The methodology, which is developed in this paper for the use in the airline industry under parametric uncertainty of airline customer demand models, may be found to be useful in other industries such as hotels, car rental companies, shipping companies, etc. While the details of problems considered in the paper can change significantly from one industry to the next, the focus is always on making better demand decisions − and not manually with guess work and intuition − but rather scientifically with models and technology, all implemented with disciplined processes and systems.
The methodology described here can be extended in several different directions to handle various problems that arise in practice. We have illustrated the proposed methodology for scale distributions(such as the exponential distribution). Application to other distributions could follow directly.
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